ON A UNIQUENESS THEOREM FOR THE FRANKLIN SYSTEM

K [email protected]

Received N/A; revised N/A; accepted N/A

This work is licensed under Creative Commons Attribution–NonCommercial International License (CC BY-NC 4.0).

In this paper we prove that there exist a nontrivial Franklin series and asequenceMnsuch that the partial sumsSMn(x)of that series converge to 0almost everywhere andλ·mes{x: supn|SMn(x)|>λ}→0 asλ→+∞.Thisshows that the boundedness assumption of the ratioMn+1Mn, used for the proofsof uniqueness theorems in earlier papers, can not be omitted

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